Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 60

come at the expense of V and the particle slows down in its parallel motion.

That the particle eventually bounces off the increasing magnetic field and returns
toward the equatorial plane and the other hemisphere can be seen as follows.
The magnitude of the Lorentz force is

||

B . From Fig. 2.10b it is clear
that the direction of the force is to the left, which is opposite to the particle
parallel velocity. Even at the right-hand side of the figure where the parallel
velocity has gone to zero, we see that F L =

| F L |= qV

⊥

B still has a small component
of force toward the magnetic equator due to the convergence of the magnetic
field lines. The parallel velocity of the particle will increase until the equatorial
plane is crossed. Then the axial component of the Lorentz force reverses sign
and the particle again slows down and is eventually reflected. If the initial pitch
angle is too small, the particle will penetrate so deeply into the atmosphere
at the end of the “bottle” that it will be lost by collisions. A “loss cone” then
develops in the distribution function such that particles with pitch angles less than
a certain value

q V

⊥ ×

α 0 escape and particles with α>α 0 are trapped for many bounce

cycles.
This bounce motion leads to another time scale with a time constant τ bounce ,

which can be considered in the particle dynamics. A second adiabatic invariant
is related to this dynamical time scale. The second invariant holds only for time
variations much longer than τ bounce . Finally, the gradient and curvature drifts

take particles entirely around the earth, creating another time scale

τ drift that is
related to a third adiabatic invariant. This has an even longer time scale, so it is
the easiest condition to break down. In general, we have

τ drift τ bounce τ gyration

and the particles gyrate many times between mirror points and bounce between
the mirrors many times, while the gradient and curvature drifts move them
around the earth. Sketches of these three oscillatory motions are presented in
Fig. 2.11. Neglecting scattering and plasma instabilities, the particles could be

Gyro motion

Bounce
motion

Drift motion
(e 2 direction)

Figure 2.11 The three oscillatory motions in the earth's magnetic field that are associated
with the three adiabatic invariants.